which implies replacing the excitation operator \(\hat{\mathcal {D}}^\dagger _{{q}_s{\epsilon }_s}(t_1^{\prime })\hat{\mathcal {D}}_{{q}_i{\epsilon }_i}(t_1) \)  with a spin-flip operator \(\hat{S}^+(t_1) \) . This equivalence is analogous to the ultrashort core-hole lifetime (UCL) approximation in equilibrium [1]}, [2]}, and it can be proven that Eq. (REF ) becomes exact in the \(\tau _{\rm core} \rightarrow 0\)  limit.
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